Optimal. Leaf size=29 \[ \frac {\log (\sin (c+d x))}{a d}-\frac {\sin (c+d x)}{a d} \]
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Rubi [A] time = 0.08, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2836, 12, 43} \[ \frac {\log (\sin (c+d x))}{a d}-\frac {\sin (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a (a-x)}{x} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a-x}{x} \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-1+\frac {a}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac {\log (\sin (c+d x))}{a d}-\frac {\sin (c+d x)}{a d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 23, normalized size = 0.79 \[ \frac {\log (\sin (c+d x))-\sin (c+d x)}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 25, normalized size = 0.86 \[ \frac {\log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - \sin \left (d x + c\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 28, normalized size = 0.97 \[ \frac {\frac {\log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {\sin \left (d x + c\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 33, normalized size = 1.14 \[ -\frac {1}{a d \csc \left (d x +c \right )}-\frac {\ln \left (\csc \left (d x +c \right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 27, normalized size = 0.93 \[ \frac {\frac {\log \left (\sin \left (d x + c\right )\right )}{a} - \frac {\sin \left (d x + c\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.76, size = 71, normalized size = 2.45 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}-\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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